**Definition.** A group $G$ is called *complete* (resp. *non-topologizable*) if each Hausdorff group topology on $G$ is Raikov-complete (resp. discrete). It is clear that each non-topologizable group is complete.

**Question 1.** Does there exist a complete topologizable group?

In particular:

**Question 2.** Is the group $SO(3,\mathbb R)$ complete?

**Question 3.** Is the group $Sym(\mathbb N)$ complete?

A simple Baire category argument shows that each complete topologizable group is uncountable.

**Remark.** There are many examples of Polish groups admitting a unique $\omega$-narrow Hausdorff group topology (so, each $\omega$-narrow Hausdorff group topology on such a group is complete), see http://www.math.uiuc.edu/~ssolecki/papers/AutomaticContinuity13.pdf.

In particular, $Sym(\mathbb N)$ is such a group.